Consider a causal LTI system S whose input x(t) and output y(t) are related by the differential equation
(a) Show that
and express the constants A, B, C, D, and E in terms of the constants a0, a1, a2, b0, b1, and b2.
(b) Show that S may be considered a cascade connection of the following two causal LTI systems:
(c) Draw a block diagram representation of S1
(d) Draw a block diagram representation of S2.
(e) Draw a block diagram representation of S as a cascade connection of the block diagram representation of S1 followed by the block diagram representation of S2.
(f) Draw a block diagram representation of S as a cascade connection of the block diagram representation of S2 followed by the block diagram representation of S1.
(g) Show that the four integrators in your answer to part (f) may be collapsed into two. The resulting block diagram is referred to as a Direct Form II realization of S, while the block diagrams obtained in parts (e) and (f) are referred to as Direct Form I realizations of S.
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